Assuming that by 'add "@{ $array_ref } to @{ $array_ref_b }"' you mean append the contents of the array referenced by $array_ref_b to the end of the array referenced by $array_ref, you need only use push.

push @{ $array_ref }, @{ $array_ref_b };

Of course TIMTOWTDI, but this is probably the most concise. Incidentally, if you wish you can save some typing by referring to the array pointed to by $array_ref with @$array_ref. It's all in perlref and perlreftut.